In the framework of Differential Geometry the trajectory
curve, integral of any n-dimensional
dynamical system may be considered as curve
in Euclidean n-space having local
metrics properties of curvatures.

The Flow Curvature Method is based on the
idea that if it is generally impossible to have a closed form of trajectory
curve it still possible to analytically compute its curvatures since it only involves its time derivatives.

The location of the points where the curvature of the trajectory curve vanishes defines a manifold called: flow curvature manifold. Since such
manifold is defined starting from the time derivatives of the velocity vector
field and so, contains information about the dynamics of the system, its only
knowledge enables to find again the main features of the dynamical system
studied and considered as the foundations of Dynamical Systems Theory. There are four of them: invariant sets, local bifurcations, slow-fast
dynamical systems and integrability (fixed
points and their stability, invariant manifolds, center manifold approximation, normal
forms, slow invariant manifold
analytical equation, first integrals of
any n-dimensional dynamical systems).

Since all the main features of Dynamical Systems Theory may be found again according to the Flow Curvature Method, i.e., starting
from the flow curvature manifold both
Dynamical Systems Theory and Flow Curvature Method are consistent and
so Flow Curvature Method represents
an alternative geometric approach for the study of dynamical systems which may
be applied to autonomous as well as non-autonomous
n-dimensional dynamical systems.

Publications NEW: All these publications are now in OPEN ACCESS. Just click on the
title

This gallery proposes slow manifolds of singularly or non-singularly perturbed dynamical systems of dimension 3, 4, 5,
6… and non-autonomous dynamical
systems as well as Mathematica Files which have
enabled to plot them.

In Slow Invariant Manifolds as
Curvature of the Flow of Dynamical Systems, the approach established in
Differential
Geometry and Mechanics Applications to Chaotic Dynamical Systems, has
been generalized to n-dimensional
dynamical systems. It has been demonstrated that the curvature of the flow, i.e., the curvature of the trajectory curves
directly provides the slow manifold
analytical equation of n-dimensional
slow-fast autonomous dynamical systems. Thus, the flow curvature method has enabled to obtain slow manifolds of many models singularly
perturbed or not of dimension 4, 5 & 6… Moreover, the flow curvature manifold invariance has
been demonstrated while using the concept of invariant manifolds introduced by G. Darboux
in 1878.

In Invariant
Manifolds of Complex Systems,localinvariance of the flowcurvaturemanifold analytical equation has been established in the case of
complex systems. Moreover, it has been demonstrated, under certain assumptions,
that such manifold is a local first integral.

This book aims
studying the French scientific contribution to the process of developing
the mathematical theory of nonlinear oscillations, particularly in the
period between the wars.

It is shown
that, contrary to what is often written, this contribution appears to be of
great importance, both through the work of French scholars caught in all
their diversity (mathematicians, physicists, engineers) and the role of a
real crossroads Science then played by France.

However, to
understand this situation, the period of this study has been
chronologically extended to the period before the First World War. The
mainstreaming of diverse literature sources such as periodicals on
electricity has then allowed to highlight a very
important text of Henri Poincaré from
1908, remained unknown. In this work, he applied the concept of limit
cycle, introduced in 1882
in his own works, to study the stability of the
oscillations of a device for radio engineering. The “discovery”
of this text led in particular to modify the classical point of view of the
historiography, which hitherto attributed to the Russian mathematician Andronov credit for having established this
correspondence, in 1929.
In this text of Poincaré,
as in most of those who were the subject of this work, there appears a strong
interaction between science and technology or, more precisely, between
mathematical analysis and radio engineering. This feature is one of the
main components of the process of developing the theory of nonlinear
oscillations.